Question

The value of the determinant $\left| \begin{matrix} 15! & 16! & 17! \\\ 16! & 17! & 18! \\\ 17! & 18! & 19! \\\ \end{matrix} \right|$ is equal to

• A. $15!+16!$
• B. $2(15!)(16!)(17!)$
• C. $15!+16!+17!$
• D. $16!+17!$

Answer 1

Answer: (B)

$2(15!)(16!)(17!)$

Answer 2

$\left| \begin{matrix} 15! & 16! & 17! \\\ 16! & 17! & 18! \\\ 17! & 18! & 19! \\\ \end{matrix} \right|$
$=(15!)(16!)(17!)\left| \begin{matrix} 1 & 16 & 17.16 \\\ 1 & 17 & 18.17 \\\ 1 & 18 & 19.18 \\\ \end{matrix} \right|$
$=(15!)(16!)(17!)\left| \begin{matrix} 1 & 16 & 272 \\\ 1 & 17 & 306 \\\ 1 & 18 & 342 \\\ \end{matrix} \right|$
$=(15!)(16!)(17!)\left| \begin{matrix} 0 & 2 & 70 \\\ 0 & 1 & 36 \\\ 1 & 18 & 342 \\\ \end{matrix} \right|$
$=(15!)(16!)(17!)[1-(72-70)]$
$=2(15!)(16!)(17!)$